5 HJB¶
5.1 Post Technology Jump¶
Controls:
\(i_k\) is a potential value for \(\frac{I_t^k}{K_t}\)
\(h_k\) is the distortion to capital accumulation.
State: - \(k\) is a realization of \(\log K\).
We have HJB for post technology jump as follows
In case of post damage jump, we call obtained solution as \(v \doteq \Phi^{\ell,II}\). And in case of pre damage jump, we call obtained solution as \(v \doteq\Phi^{II}\).
First order condition¶
Let
\[m u \doteq \delta\left(\frac{\alpha-i_k}{\exp (v)} \exp (k)\right)^{-\rho} \frac{\exp (k)}{\exp (\Phi^{m,II})}\]
Then first order condition for \(i_k\) is
\[m u=\frac{d \Phi^{\ell,II}}{dk}\left(1-\kappa i_k\right)\]
while first order condition for \(h_k\) is
\[\frac{d \Phi^{m,II}}{dk} \sigma_k=-\xi_k h_k\]
5.2 Pre technology Jump¶
5.2.1 Pre Technology and Post Damage Jump¶
Controls: - \(i_k\) is a potential value for \(\frac{I_t^k}{K_t}\) - \(i_j\) is a potential value for \(\frac{I_t^j}{K_t}\) - \(\mathcal{E}\) is a potential value for \(\mathcal{E}_t\) - \(h_k\) is the distortion to capital accumulation. - \(h_y\) is the distortion to temperature anomaly accumulation. - \(h_j\) is the distortion to R&D accumulation.
State: - \(k\) is a realization of \(\log K\). - \(y\) is a realization of \(Y\). - \(j\) is a realization of \(\log J\). - \(n\) is a realization of \(\log N\).
We attempt to solve a value function of the form
\[V^\ell(k,y,j,n) = \frac{\partial V^\ell}{\partial n} n + \Phi^\ell(k, y, j)\]
5.2.1 Pre Technology and Pre Damage Jump¶
Controls: - \(i_k\) is a potential value for \(\frac{I_t^k}{K_t}\) - \(i_j\) is a potential value for \(\frac{I_t^j}{K_t}\) - \(\mathcal{E}\) is a potential value for \(\mathcal{E}_t\) - \(h_k\) is the distortion to capital accumulation. - \(h_y\) is the distortion to temperature anomaly accumulation. - \(h_j\) is the distortion to R&D accumulation.
State: - \(k\) is a realization of \(\log K\). - \(y\) is a realization of \(Y\). - \(j\) is a realization of \(\log J\). - \(n\) is a realization of \(\log N\).
We attempt to solve a value function of the form
\[V(k,y,j,n) = \frac{\partial V}{\partial n} n + \Phi(k, y, j)\]
[1]:
from src.plot import plot_simulatedpath_full
plot_simulatedpath_full(graph_type="RD_Plot", graph_title = "Figure 2: R&D Investment as Percentage of GDP", yaxis_label="%", graph_range=[0,10], before15=True, abatement_cost = 0.5, rho=0.66)
[3]:
from src.plot import plot_simulatedpath_uncer_decomp
plot_simulatedpath_uncer_decomp(graph_type="RD_Plot", graph_title = "2: R&D Investment as Percentage of GDP", yaxis_label="%", graph_range=[0,10], before15=True, abatement_cost = 0.1, rho=0.66)
[1]:
from src.plot import plot_climatehist
plot_climatehist()
[2]:
from src.plot import plot_gammahist
plot_gammahist("Figure 6: Distorted climate model distribution ")
[1]:
from src.plot import plot_simulatedpath_full2
plot_simulatedpath_full2(graph_type="RD_Plot", graph_title = "Figure 2: R&D Investment as Percentage of GDP", yaxis_label="%", graph_range=[0,10], before15=True)
[2]:
from src.plot import plot_simulatedpath_uncer_decomp2
plot_simulatedpath_uncer_decomp2(graph_type="RD_Plot", graph_title = "Figure 2: R&D Investment as Percentage of GDP", yaxis_label="%", graph_range=[0,10], before15=True)
[3]:
import plotly.io as pio
import plotly.offline as pyo
import pandas as pd
import numpy as np
from plotly.subplots import make_subplots
import plotly.graph_objects as go
import os
import sys
import pickle
sys.path.append(os.path.dirname(os.getcwd()))
pyo.init_notebook_mode()
pio.templates.default = "none"
theta_list = pd.read_csv('data/model144.csv', header=None).to_numpy()[:, 0] / 1000.
gamma_3_list = np.linspace(0, 1./3, 20)
def plot_simulatedpath_uncer_decomp(graph_type, graph_title, yaxis_label, graph_range, before15, abatement_cost, rho):
xi_base = 100000.
psi_0 = 0.10583
psi_1 = 0.5
varrho = 1120.0
delta=0.01
# fig = make_subplots(1, 2)
fig = go.Figure()
color = ["#d62728", "darkgreen", "darkorange", "navy"]
for abatement_cost in [0.1, 0.5]:
for rho in [0.66, 1.0, 1.5]:
folder = "./data_simul2/2jump_step_4.00,9.00_0.0,4.0_1.0,6.0_0.0,3.0_SS_0.2,0.1,0.1_LR_0.0025_FK_phi0_{}/".format(
abatement_cost)
for xi_num in range(4):
xi_list_uncertaintydecomp = 100000.0 * np.ones((5))
if abatement_cost==0.1:
xi_list_uncertaintydecomp[xi_num] = 0.050
if xi_num==2:
xi_list_uncertaintydecomp[4] = 0.050
if abatement_cost == 0.5:
xi_list_uncertaintydecomp[xi_num] = 0.150
if xi_num==2:
xi_list_uncertaintydecomp[4] = 0.150
# print(xi_list_uncertaintydecomp)
# filename = "xi_a_{}_xi_k_{}_xi_c_{}_xi_j_{}_xi_d_{}_xi_g_{}_psi_0_{}_psi_1_{}_varrho_{}_rho_{}_delta_{}_" .format(
# xi_base, xi_list_uncertaintydecomp[0], xi_list_uncertaintydecomp[1], xi_list_uncertaintydecomp[2], xi_list_uncertaintydecomp[3],xi_list_uncertaintydecomp[4], psi_0, psi_1, varrho, rho, delta)
filename = "xi_a_{}_xi_k_{}_xi_c_{}_xi_j_{}_xi_d_{}_xi_g_{}_psi_0_{}_psi_1_{}_varrho_{}_rho_{}_delta_{}_" .format(
xi_base, xi_list_uncertaintydecomp[3], xi_list_uncertaintydecomp[0], xi_list_uncertaintydecomp[2], xi_list_uncertaintydecomp[1],xi_list_uncertaintydecomp[4], psi_0, psi_1, varrho, rho, delta)
with open(folder + filename + "model_tech1_pre_damage_UD_simul_40direct_direct", "rb") as f:
model_tech1_pre_damage = pickle.load(f)
# print(model_tech1_pre_damage.keys())
# print(filename)
# label = r'ξᵣ = {:.1f}'.format(xi_list_fullaversion[i])
if xi_num == 0:
label = "Climate Uncertainty"
if xi_num == 1:
label = "Damage Uncertainty"
if xi_num == 2:
label = "Technology Uncertainty"
if xi_num == 3:
label = "Producticity Uncertainty"
# print(model_tech1_pre_damage[graph_type])
if before15==False:
fig.add_trace(go.Scatter(x=model_tech1_pre_damage["years"],
y=model_tech1_pre_damage[graph_type],
name=label,
showline=False,
showlegend=False,
line=dict(color=color[xi_num]),
# visible=False
))
elif before15==True:
fig.add_trace(go.Scatter(x=model_tech1_pre_damage["years"][model_tech1_pre_damage["states"][:, 1]<1.5],
y=model_tech1_pre_damage[graph_type][model_tech1_pre_damage["states"][:, 1]<1.5],
name=label,
showline=False,
showlegend=False,
line=dict(color=color[xi_num]),
# visible=False
))
for i in range(3):
fig.data[3*2 + i]["visible"] = True
# fig.data[3*2 + i+12]["visible"] = True
fig.data[3*2 + i]["showlegend"] = True
# fig.data[3*2 + 12]["showlegend"] = True
buttons = []
i = 0
for abatement_cost in [0.1, 0.5]:
for rho in [0.66, 1.0, 1.5]:
if abatement_cost == 0.1:
xi_list = [0.025, 0.050]
cost_label = "Low Cost"
if abatement_cost == 0.5:
xi_list = [0.075, 0.150]
cost_label = "High Cost"
# Hide all traces
label = cost_label+r', ρ' + '= {:.2f}'.format(rho)
button = dict(method='update',
args=[
{
'visible': [False] * (2 * 3 * 4 * 2),
'showlegend': [False] * (2 * 3 * 4 * 2),
},
],
label=label)
# Enable the two traces we want to see
# print(button['args'][0]["visible"])
button['args'][0]["visible"][4*i + 0] = True
button['args'][0]["visible"][4*i + 1] = True
button['args'][0]["visible"][4*i + 2] = True
button['args'][0]["visible"][4*i + 3] = True
button['args'][0]["showlegend"][4*i + 0] = True
button['args'][0]["showlegend"][4*i + 1] = True
button['args'][0]["showlegend"][4*i + 2] = True
button['args'][0]["showlegend"][4*i + 3] = True
i = i+1
# Add step to step list
buttons.append(button)
fig.update_layout(
updatemenus=[
dict(
type="buttons",
buttons=buttons,
# direction="right",
active=2,
x=1.3,
y=0.7,
# xanchor="left",
# yanchor="top",
pad={"r": 10,
"t": 10, "b": 10},
showactive=True
)
])
fig.update_xaxes(showgrid=False, showline=True,
title="Years", range=[0, 40])
fig.update_yaxes(showgrid=False,
showline=True,
range=graph_range,
title_text= yaxis_label,
tickformat=".2f")
# fig.update_yaxes(showgrid=False, showline=True,
# range=[1., 2.1], col=2, row=1)
# fig.update_yaxes(tickvals=[1, 1.2, 1.4, 1.5, 1.6, 1.8, 2.0],
# tickformat=".1f",
# col=2,
# row=1)
# fig.update_layout(height=400, width=1280)
fig.update_layout(
title=graph_title,
barmode="overlay",
plot_bgcolor="white",
width=800,
height=600,
margin=dict(l=50, r=0))
return fig
[2]:
# from src.plot import plot_simulatedpath_uncer_decomp2
plot_simulatedpath_uncer_decomp2(graph_type="RD_Plot", graph_title = "Figure 11: R&D Investment as Percentage of GDP", yaxis_label="%", graph_range=[0,10], before15=True)
[ ]: